# Based on continuous condition attributes reduction algorithm fuzzy rules

Abstract: For continuous domains rough sets attribute decision table is not easy to deal with poor ability to obtain the relationship between fuzzy sets and other issues, propose a fuzzy sets and rough sets combine continuous condition attribute reduction algorithm of fuzzy rules of the Firstly, the introduction of triangular membership function continuous attribute values ​​into fuzzy values, and the use of discrete fuzzy neural network method to obtain the relationship between data sets. instance verification shows that this algorithm can be based on the actual decision-making needs of users and domain knowledge to change the thresholds, fuzzy rules to obtain a satisfactory result.

Keywords: condition attribute; continuous; membership function; fuzzy rules

## Attribute reduction algorithms of fuzzy rules based on? Continuous domain condition attributes

CUI Meng-tian? 1, ZHU Hao-dong? 2, ZHONG Yong? 2? (1.School of Computer Science & Technology, Southwest University for Nationalities, Chengdu 610041, China; 2.Chengdu Institute of? Computer Applications, Chinese Academy of Sciences, Chengdu610041, China)
Abstract: To solve the problems of low adaptability for continuous domain reduction and the disadvantage of failing to obtain eventual relationship among the fuzzy sets, this paper proposed a new method of attribute reduction algorithms of decision table based on combining fuzzy set with rough set. First , transformed continuous attribute value into fuzzy value with triangular membership function, then provided algorithms of hard C-means (HCM) clustering to obtain relationship among the fuzzy sets.In the end, simulation results show the effectiveness of the proposed method through an illustrative example .

## 0 Introduction

Rough set theory [1] is an investigational imprecise, uncertain mathematical tool, its main advantage is that thinking and the ability to maintain the same classification under the premise of knowledge reduction through export issues decisions or classification rules. The theory attributes reduction is a very important concept, it reflects the nature of a decision table information, has been widely used [2].

In reality, most of the data set attribute values ​​are continuous type. These continuous data mostly with strong ambiguity, the boundaries between the concepts are not very clear because of the traditional rough set theory is very suitable for processing discrete domain properties decision table, for the continuous domain attribute decision table processing capability is very limited, which greatly limits its application if the rough set theory is applied to the continuity property, then prior to use of the theory of continuous attributes must be discretized. However, the attribute value after discrete attribute values ​​not retained on real values ​​existing differences, which will result in some degree of information loss. Therefore, rough set theory needs to deal with other problems imprecise or uncertain Combing in order to expand its scope of application.

Fuzzy set theory is a kind used in modeling the experimental data for some of the uncertainty and ambiguity issues a powerful tool of its advantages: fuzzy set theory provides a systematic, language computing tools represent such information, expressed by the membership function by using linguistic variables, it can be calculated. reasonable choice of fuzzy rules is a key factor in the fuzzy inference system, which can effectively for specific applications in the field of human expertise modeled. Pawlak rough set point theory and fuzzy set theory are not mutually exclusive but can complement each other [3]; Dubois et al [4] further indicated that they are uncertain knowledge processing two mathematical methods are complementary in nature. To this end, This paper proposes a rough sets and fuzzy sets combine continuous condition attribute reduction algorithm of fuzzy rules.

## A related definitions

In order to better describe the algorithm, first some definitions are given below for? Bedding.
Defines a continuous domain decision table S = <U,C,D,V,f>. Wherein: U is non-empty finite set of objects U = {u? 1, u? 2, ..., u? N}; C = {c ? 1, c? 2, ..., c? m} is a set of condition attributes, each attribute is continuous attributes; D = {d} is the decision attribute.
For? C? J ∈ C (j = 1,2, ..., m), the membership function can be used to its continuous attribute values ​​into fuzzy values. Use I? J? K represent continuous attributes c? J's k-th fuzzy interval, m? j represents c? j fuzzy interval number, μ? kij represent the object u? i (i = 1,2, ..., n) in the fuzzy interval I? j? k membership, vij represents u? i in c? j of attribute values ​​vij can be expressed as follows:

vij = μ? 1ij / I? j? 1 + μ? 2ij / I? j? 2 + ... + μ?? m?? j? ij / I?? m?? j?? j (1)
Definition 2 For continuous domain decision table S = <U,C,D,V,f>, object u? I and u? S in continuous attribute c? J similarity is defined as follows:

## μc?? j (u? i, u? s) = 1-1m? j? m? jt = 1 | μ? tij-μ? 1sj | (2)

Definition 3 For continuous domain decision table S = <U,C,D,V,f>, object u? I in a continuous-type attribute c? J on a similar class defined as follows:
sim? βc?? j (u? i) = {u? t | μc?? j (u? i, u? t)> = β, t = 1,2, ..., n} (3)
Where: β is given the similarity threshold.
Definition 4 For continuous domain decision table S = <U,C,D,V,f>, continuous attribute c? J divided on U formed clusters consisting of similar vectors are defined as follows:

simClassVector (c? j) = (sim? βc?? j (u? i) | i = 1,2, ..., n) (4)

## Two figures similar feature vectors and matrices

In the decision table, each attribute that their characteristics can be found in a vector, the vector can be called a number of properties characteristic vector.

Definition 5 For continuous domain decision table S = <U,C,D,V,f>, assuming continuous attribute c? I on U division formed similar clusters composed of vector is defined as simClassVector (c? J) = (sim? βc?? j (u? i) | i = 1,2, ..., n), then the continuous attribute c? i feature vector is defined as the number

DCV (c? I) = (λit | λit = card (sim? Βc?? I (u? T)), t = 1,2, ..., n) (5)

fuzzy set theory is based on fuzzy relations, the easiest way is similar to the relationship between the performance. Similarity relation is reflexive and symmetric binary fuzzy relation. Similar relationships can be constructed of many similar matrix similarity matrix transitive closure is fuzzy equivalence relation, each of λ cut sets are usual sense equivalence relation.

Definition 6 For continuous domain decision table S = <U,C,D,V,f>, continuous attributes c? I (i = 1,2, ..., m) digital eigenvectors DCV (c? I), continuous attributes is defined as the similarity between the matrix [R] = (rij) m × n. wherein [R] is defined as each element

rij = 1-δ ×? nk = 1 | λik-λjk | (6)
Where: i, j = 1,2, ..., m; 0 <δ <1 is a constant; m is the total number of condition attributes.

## 3 new attribute reduction algorithm

The proposed new attribute reduction algorithm is suitable for the condition attribute is continuous decision-making table, which is described as follows:

Input: continuous domain decision table S = <U,C,D,V,f>, similarity threshold β, similar to the matrix element constant coefficient δ, fuzzy equivalent matrix cut set threshold λ.

Output: the satisfaction of subjective conditions set attribute reduction and fuzzy rule sets.

## e) Select the appropriate threshold λ, to obtain a satisfactory subjective conditions attribute? Jane sets;

f) According to the subjective conditions set attribute reduction, export the appropriate fuzzy rule set, the algorithm terminates.

## 4 Examples

In this paper, the diesel engine fuel injection system fault diagnosis, for example, the data in Table 1 is formed by a decision table Troubleshooting [3,5], where: u? 1, u? 2, ..., u? 6 denote the six systems status; c? 1, c? 2, c? 3 of condition attributes represent stable precision repair operation fix accuracy, robustness degree; d is the decision attribute, which means that repair effect.

## u? 64020

According to the literature [6,7] provides conditional attribute segmentation method, and literature [8,9] provided by the triangular membership function, each divided into five fuzzy interval continuous attributes, including those attributes do not appear fuzzy fuzzy interval is not in the table said came out, finally get the system of fuzzy decision table.

Here take β = 0.8 calculated for each individual under conditions similar to class attributes.

## Calculated c? 1 under each category is similar to

sim?? 0.8c?? 1 (u? 1) = {u? 1}
sim?? 0.8c?? 1 (u? 2) = {u? 2, u? 4, u? 5}
sim?? 0.8c?? 1 (u? 3) = {u? 3, u? 5, u? 6}
sim?? 0.8c?? 1 (u? 4) = {u? 2, u? 4}
sim?? 0.8c?? 1 (u? 5) = {u? 2, u? 3, u? 5, u? 6}
sim?? 0.8c?? 1 (u? 2) = {u? 3, u? 5, u? 6}
Therefore, DCV (c? 1) = (3,4,2,4,, 3,1). Similarly available DCV (c? 2) = (1,3,2,3,4,3), DCV (c? 3) = (2,3,3,3,3,2).

Numbers by each condition attribute feature vector, take δ = 0.02, using the fuzzy matrix closure operation method [9,10] can be obtained
[T (R)] = 10.560.56? 0.5610.56? 0.560.561
Take λ = 0.8 can be obtained

## [T (R)]? Λ = 1 0 0? 0 1 0? 0 0 1

The cut in the fuzzy equivalent matrix set threshold value λ = 0.8 under the condition of each successive condition attribute is not relevant and therefore in Table 1 is a subjective reduction set {c? 1, c? 2, c? 3}, this result with the literature [8] the results exactly.

Through this example illustrates the use of the algorithm can not only solve the continuous domain decision table attribute reduction issues, but also may need to obtain a subjective attribute reduction set and a set of fuzzy rule sets, which illustrate the algorithm is feasible.

## 5 Conclusion

In this paper, rough set for continuous processing domain attributes poor decision table, and not easy to obtain the relationship between fuzzy sets and other issues, propose a fuzzy sets and rough sets to combine continuous condition attribute reduction algorithm of fuzzy rules. Instance validation show that using this algorithm, the user can according to the actual decision-making needs and domain knowledge to change the threshold to obtain a satisfactory result of fuzzy rules.

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